Constructing the algebras D and the involutions . To a fake - PowerPoint PPT Presentation

Constructing the algebras D and the involutions ι . To a fake projective plane there is associated a pair ( k, ℓ ) of fields coming from a short list. There is also an algebra D , an involution ι and a group G , with G ( k ) = { ξ ∈ D : ι ( ξ ) ξ = 1 & Nrd( ξ ) = 1 } . D , ι and G must satisfy the properties: 1

• G ( k v ) ∼ = SL (3 , k v ) for all v ∈ V f \ T 0 which split in ℓ , • G ( k v ) ∼ = { g ∈ SL (3 , k v ( s )) : g ∗ F v g = F v } if v ∈ V f does not split in ℓ , • G ( k v ) is compact for v ∈ T 0 , • G ( k v ) ∼ = SU (2 , 1) for one archimedean place v on k , and • G ( k v ) ∼ = SU (3) for the other archimedean place v on k (if k � = Q ). We know that T 0 = ∅ if D = M 3 × 3 ( ℓ ), and that T 0 is a specific singleton if D is a division algebra. 2

Corollary 6.6 in Chapter 10 of W. Scharlau “Quadratic and Hermitian Forms”: The above properties determine D and ι up to k -isomorphism or anti- isomorphism. Anti-isomorphism must be allowed here because given D , we can define an “opposite” algebra D op whose elements x op are in 1-1 correspondence ( x op ↔ x ) with those of D , and in which for all x, y ∈ D and t ∈ ℓ , x op + y op = ( x + y ) op , tx op = ( tx ) op , and x op y op = ( yx ) op . 3

Suppose that ℓ is a field and that m is a Galois extension of ℓ of degree 3, with Gal( m/ℓ ) = � ϕ � . Fix some nonzero D ∈ ℓ , and form D = { a + bσ + cσ 2 : a, b, c ∈ m } , which we can make into an associative algebra of dimension 9 over ℓ in which σ 3 = D and σa = ϕ ( a ) σ for all a ∈ m. The centre of D is ℓ . We shall see in a moment that D has no non-trivial two-sided ideals — D is a central simple algebra. 4

There is an ℓ -algebra homomorphism Ψ : D → M 3 × 3 ( m ) such that     a 0 0 0 1 0 0 ϕ ( a ) 0 Ψ( σ ) = 0 0 1 and Ψ( a ) =  .        ϕ 2 ( a ) D 0 0 0 0 So if ξ = a + bσ + cσ 2 ∈ D , then   a b c Dϕ ( c ) ϕ ( a ) ϕ ( b ) Ψ( ξ ) =  .    Dϕ 2 ( b ) Dϕ 2 ( c ) ϕ 2 ( a ) 5

The reduced norm Nrd( ξ ) of ξ is det(Ψ( ξ )) = aϕ ( a ) ϕ 2 ( a ) + Dbϕ ( b ) ϕ 2 ( b ) + D 2 cϕ ( c ) ϕ 2 ( c ) − D ( aϕ ( b ) ϕ 2 ( c ) + ϕ ( a ) ϕ 2 ( b ) c + ϕ 2 ( a ) bϕ ( c )) . Then Nrd : D → ℓ , and Nrd( ξη ) = Nrd( ξ )Nrd( η ) for all ξ, η ∈ D . An element ξ = a + bσ + cσ 2 of D is invertible if and only if Nrd( ξ ) � = 0, in which case ξ − 1 equals 1 �� � ϕ ( a ) ϕ 2 ( a ) − Dϕ ( b ) ϕ 2 ( c ) � � Dcϕ 2 ( c ) − bϕ 2 ( a ) � � � σ 2 + σ + bϕ ( b ) − cϕ ( a ) . Nrd( ξ ) 6

Proposition. Either (a) D ∼ = M 3 × 3 ( ℓ ), or (b) D is a division algebra. Case (a) holds if and only if D is the norm N m/ℓ ( η ) of an element η of m . Proof. If D = N m/ℓ ( η ), let   0 0 η C = 0 1 0  .    0 0 1 /ϕ ( η ) 7

Let θ 0 , θ 1 and θ 3 be basis for m over ℓ . Form ϕ 2 ( θ 0 )   ϕ ( θ 0 ) θ 0 ϕ 2 ( θ 1 ) Θ = θ 1 ϕ ( θ 1 )  .    ϕ 2 ( θ 2 ) θ 2 ϕ ( θ 2 ) Then   ζ 0 ζ 1 ζ 2 Θ − 1 = ϕ ( ζ 0 ) ϕ ( ζ 1 ) ϕ ( ζ 2 )  ,    ϕ 2 ( ζ 0 ) ϕ 2 ( ζ 1 ) ϕ 2 ( ζ 2 ) where Trace( θ i ζ j ) = δ ij . J := Θ C . Then J Ψ( ξ ) J − 1 has entries in ℓ . E.g., ( J Ψ( σ ) J − 1 ) ij = Trace( θ i ηϕ ( ζ j )). So ξ �→ J Ψ( ξ ) J − 1 is a ℓ -linear algebra homomomorphism D → M 3 × 3 ( ℓ ). It is clearly injective, and so an isomorphism, as dimensions match. 8

If D is not equal to N m/ℓ ( η ) for any η ∈ m , then Nrd(1 + cσ 2 ) = 1 + D 2 N m/ℓ ( c ) Nrd(1 + bσ ) = 1 + DN m/ℓ ( b ) and cannot be zero for any b, c ∈ m . So any 1 + bσ or 1 + cσ 2 is invertible. So any nonzero element of D is invertible, and D is a division algebra. Corollary. D is a central simple algebra over ℓ . 9

Suppose ℓ = k ( s ), where s 2 = − κ ∈ k . We want an involution ι of the second kind on D . Assume m normal extension of k . The conjugation automorphism ex- tends to m . Then either ϕ ( a ) = ϕ 2 (¯ ϕ ( a ) = ϕ (¯ a ) for all a ∈ m OR a ) for all a ∈ m. Gal( m/k ) is abelian in the first case, and non-abelian in the second case. 10

Lemma. If Gal( m/k ) is non-abelian, and if D ∈ ℓ satisfies D = D � = 0, then there is an involution of the second kind ι : D → D such that ι ( σ ) = σ and ι ( a ) = ¯ a for all a ∈ m. Explicitly, ι ( a + bσ + cσ 2 ) = ¯ b ) σ + ϕ 2 (¯ c ) σ 2 . a + ϕ (¯ Note that   D 0 0 Ψ( ι ( ξ )) = F − 1 Ψ( ξ ) ∗ F for F = 0 0 1  .    0 1 0 11

In each of the five k = Q cases ( a = 1 , p = 5),...,( a = 23 , p = 2), we can choose a field m as above, with Gal( m/k ) non-abelian, and define D using D = p . In each of these cases, there is a β ∈ ℓ so that ¯ ββ = 2 p . Then   β 0 0 F = 1 2∆ ∗ F 0 ∆ for ∆ = 0 1 1  .    − 1 0 1 If ι ( ξ ) ξ = 1 then (∆Ψ( ξ )∆ − 1 ) ∗ F 0 (∆Ψ( ξ )∆ − 1 ) = F 0 . 12

So if G ( k ) = { ξ ∈ D : ι ( ξ ) ξ = 1 and Nrd( ξ ) = 1 } , then ξ �→ ∆Ψ( ξ )∆ − 1 defines an injective homomorphism G ( k ) → SU (2 , 1). In fact, G ( k v ) ∼ = G ( R ) ∼ = SU (2 , 1) for the one archimedean place v on k = Q — see below. We can take β = 3 + i in the case ( a = 1 , p = 5), β = 2 + s for ( a = 2 , p = 3) and β = 2 for the other three cases. 13

In the 5 cases ( k, ℓ ) in which k = Q and ℓ = Q ( s ), we can define m = Q ( s, Z ), where Z satisfies P ( Z ) = 0 for a cubic monic P ( X ) ∈ Z [ X ]. s 2 p P ( X ) ϕ ( Z ) X 3 − 3 X 2 − 2 ( s + 3 − (4 s + 1) Z + sZ 2 ) / 2 − 1 5 X 3 + X 2 + 2 X − 2 (2( s − 1) + (3 s − 2) Z + sZ 2 ) / 4 − 2 3 X 3 + 3 X 2 + 3 − (3( s + 7) + (9 s + 7) Z + 2 sZ 2 ) / 14 − 7 2 X 3 − 3 X − 3 (4 s + (3 s − 5) Z − 2 sZ 2 ) / 10 − 15 2 X 3 − X − 1 (4 s + (9 s − 23) Z − 6 sZ 2 ) / 46 − 23 2 In each case Gal( m/ℓ ) = � ϕ � , and Gal( m/ Q ) is non-abelian. In the case ( a = 7 , p = 2), we shall use a different cyclic simple algebra, coming from a field m so that Gal( m/k ) is abelian. 14

Lemma. If Gal( m/k ) is abelian, and if D ∈ ℓ satisfies ¯ DD = 1, then there is an involution ι 0 : D → D of the second kind such that ι 0 ( σ ) = σ − 1 and ι 0 ( a ) = ¯ for all a ∈ m. a Explicitly, ι 0 ( a + bσ + cσ 2 ) = ¯ Dϕ 2 (¯ b ) σ 2 . a + ¯ c ) σ + ¯ Dϕ (¯ It is easy to check that Ψ( ι 0 ( ξ )) = Ψ( ξ ) ∗ . For reasons explained on the next slide, we shall use the involution ι ( ξ ) = T − 1 ι 0 ( ξ ) T, where T ∈ m and ¯ T = T � = 0. Then   0 0 T Ψ( ι ( ξ )) = F − 1 Ψ( ξ ) ∗ F 0 ϕ ( T ) 0 for F =  .    ϕ 2 ( T ) 0 0 15

Embedding m in C , the images of T , ϕ ( T ) and ϕ 2 ( T ) are real because ϕ ( T ) = ϕ ( ¯ T ) = ϕ ( T ). If T > 0, ϕ ( T ) > 0 and ϕ 2 ( T ) < 0, then   | T | 1 / 2 0 0 F = ∆ ∗ F 0 ∆ | ϕ ( T ) | 1 / 2   for ∆ =  . 0 0    | ϕ 2 ( T ) | 1 / 2 0 0 So ξ �→ ∆Ψ( ξ )∆ − 1 is an injective homomorphism G ( k ) → SU (2 , 1). In the cases ( a = 7 , p = 2), C 2 , C 10 , C 18 and C 20 , we define D = { a + bσ + cσ 2 : a, b, c ∈ m, where σ 3 = D and σxσ − 1 = ϕ ( x ) } for the following m ’s and D ’s: 16

name m v 0 D ϕ ζ 7 �→ ζ 2 ( a = 7 , p = 2) Q ( ζ 7 ) 2 (3 + s ) / 4 7 (1 + √− 15) / 4 ζ 9 �→ ζ 4 C 2 k ( ζ 9 ) 2 9 W �→ 2 − W − W 2 C 10 ℓ ( W ) 2 rU/ 2 ζ 9 �→ ζ 4 C 18 k ( ζ 9 ) 3 ( r + 1 + 2 ω ) / 3 9 (3 + √− 7) / 4 ζ 7 �→ ζ 2 C 20 k ( ζ 7 ) 2 7 In case C 2 , k = Q ( r ), where r 2 = 5 and ℓ = k ( ω ), where ω = ζ 3 , In case C 10 , k = Q ( r ), where r 2 = 2, ℓ = k ( U ), where U 2 = ( r + 1) U − 2, and W 3 − 3 W + 1 = 0. In case C 18 , k = Q ( r ), where r 2 = 6, and ℓ = k ( ω ), In case C 20 , k = Q ( r ) where r 2 = 7, and ℓ = k ( i ). 17

Having chosen m as above, with Gal( m/k ) abelian, the subfield { a ∈ m : a = a } has the form k ( W ), where W satisfies an equation Q ( W ) = 0 for ¯ some monic cubic Q ( X ) ∈ Z [ X ]. We choose T ∈ k ( W ) as follows: name W ϕ ( W ) T W 2 − 2 ζ 7 + ζ − 1 ( a = 7 , p = 2) W 7 ζ 9 + ζ − 1 2 − W − W 2 − 2 r + ( r − 1) W + 2 W 2 C 2 9 2 − W − W 2 − r + (1 − r ) W + W 2 C 10 W ζ 9 + ζ − 1 2 − W − W 2 3 − 3 r + rW + rW 2 C 18 9 W 2 − 2 ζ 7 + ζ − 1 2 + W − (4 + 3 W + W 2 ) /r C 20 7 In the first and last cases, Q ( X ) = X 3 + X 2 − 2 X − 1. In the other three cases, Q ( X ) = X 3 − 3 X + 1. 18

The above choice is made so that, in the four cases k = Q ( r ), where r 2 = N ( N = 5, 2, 6 or 7), and fixing a solution W R ∈ R of Q ( X ) = 0, √ • embedding k ( W ) in R by mapping r to + N and W to W R , the images of T , ϕ ( T ) and ϕ ( T ) DO NOT all have the same sign, and √ • embedding k ( W ) in R by mapping r to − N and W to W R , the images of T , ϕ ( T ) and ϕ ( T ) DO all have the same sign. This implies that G ( k v ) ∼ = SU (2 , 1) for the archimedean valuation v corre- sponding to the first embedding, and G ( k v ) ∼ = SU (3) for the archimedean valuation v corresponding to the second embedding. 19